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The equation (7.3.3-13) can be rewritten in a standard form of weighting spectrum
σ p + σ bi
ϕi
( u)
=
(7.3.3-16)
σ ( u)
+ σ
t
bi
σ
bi
1
= ∑ n jσ
j
n
i
j≠i
bi
+
n l
i i
, (7.3.3-17)
where n i and n j are the atomic number densities of the absorber under consideration and of admixed
moderators, respectively.
The standard spectrum of Eq.(7.3.3-16) was obtained again from the behavior of the collision
probability at
σ t → ∞ . So we need some corrections for the present approach, as done in the previous
section. For this purpose we at first define the generalized Dancoff correction factor of an absorber in
the region i by
C
= 1−
i b i
(7.3.3-18)
Then the two-region problem in the previous subsection suggests the replacement of (1 - C i ) by
(1 - C i )g(C i ) with
a
g(
Ci
) = 1 + ( a −1)
C
. (7.3.3-19)
i
Accordingly we can generally define the background cross-section, σ bi , including the
heterogeneity by
1 g(
C )(1 − C )
= ∑
i i
σ bi n jσ
j +
. (7.3.3-20)
ni
j≠i
nili
Here, the value of the Bell factor, a, of Eq.(7.3.3-19) is assumed to take the respective value of Table
7.3.2-1 corresponding to the geometry under consideration.
We again obtain the equivalence relation: The effective cross-sections of absorber nuclides in
each region can be calculated by using a cross-section set of the Bondarenko type.
In the resonance energy range I (E ≥ 130.07 eV), the effective cross-sections are obtained by the
table-look-up method of a Bondarenko type cross-section set, where the heterogeneity is treated by the
established equivalence relation of Eq.(7.3.3-20) between heterogeneous and homogeneous mixtures.
The cross-sections in the second region (130.07 eV ≥ E ≥ 0.414 eV are generally calculated by the IRA
or a direct numerical method using collision probability and hyper-fine group width (Δu = 0.00125), as
described in the next subsection. Hence, the energy range concerning the present improvement is
mainly the first resonance region.
As known through the present derivation, each resonant nuclide in one absorber lump may take
a different value for the generalized Dancoff correction factor. One example of this type of problems
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